Fourier transform apparatus

ABSTRACT

A Fourier transform apparatus includes: a signal generating section for generating a plurality of sine-wave signals and a plurality of cosine-wave signals; a plurality of analog circuits each having a respective circuit parameter corresponding to a respective Fourier coefficient, and each receiving the respective sine-wave signal and the respective cosine-wave signal which are generated by the signal generating section; and an operation section for performing an operation on each of outputs of the respective analog circuits and outputting the resultant respective analog signals.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a Fourier transform apparatus forperforming Fourier transform used for processes such as signal analysis,signal compression and decoding.

2. Description of the Related Art

Fourier transform is indispensable for the processes such as signalanalysis, signal compression and decoding. The Fourier transform isbased on the idea that “any periodic function can be represented as thesum of trigonometric functions”. A non-periodic signal is considered asa function having an infinite cycle.

Recently, discrete Fourier transform has often been used. A cycle of asample signal obtained from N sample values of t=0 to t=(N−1) is T=N. Afrequency f_(N) of this signal is given by the following expression (1):

f _(N)=1/N  (1)

A component of the frequency f_(N) is a fundamental-wave component,whereby a harmonic-wave component having a frequency k/N equal to thefrequency f_(N) multiplied by an integer can be obtained. By using thesefrequencies, the following expressions of discrete Fourier transform andinverse discrete Fourier transform can be defined based on thedefinition of the Fourier transform.

Fourier-transform expression (represented by sine-wave and cosine-wavecomponents): $\begin{matrix}\left. \begin{matrix}{{a(0)} = {\frac{1}{N}{\sum\limits_{k = 0}^{k = {N - 1}}{g(k)}}}} \\{{a\left( {n/N} \right)} = {\frac{1}{N}{\sum\limits_{k = 0}^{k = {N - 1}}{{g(k)}\cos \quad \left( {{- 2}\pi \quad {{kn}/N}} \right)}}}} \\{{b\left( {n/N} \right)} = {\frac{1}{N}{\sum\limits_{k = 0}^{k = {N - 1}}{{g(k)}\quad \sin \quad \left( {{- 2}\pi \quad {{kn}/N}} \right)}}}}\end{matrix} \right\} & (2)\end{matrix}$

Inverse Fourier-transform expression (represented by sine-wave andcosine-wave components): $\begin{matrix}{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}{{a\left( {k/N} \right)}\cos \quad \left( {2\pi \quad {{kn}/N}} \right)}} + {\sum\limits_{k = 1}^{N - 1}{{b\left( {k/N} \right)}\sin \quad \left( {2\pi \quad {{kn}/N}} \right)}}}} & (3)\end{matrix}$

where g(n) is a sample signal, and a(n/N) and b(n/N) are Fouriercoefficients.

Each of the above expressions, Expressions (2) and (3), is a Fourierexpansion expression in which the sine-wave component and cosine-wavecomponent are separated.

Moreover, each of the above Fourier expansion expressions is representedby a complex number by using the following Expression (4):

G(n/N)=a(n/N)+jb(n/N)  (4)

Fourier-transform expression (represented by a complex number):$\begin{matrix}{{G\left( {n/N} \right)} = {\frac{1}{N}{\sum\limits_{k = 0}^{k = {N - 1}}{{g(k)}\exp \quad \left( {{- j}\quad 2\pi \quad {{nk}/N}} \right)}}}} & (5)\end{matrix}$

Inverse Fourier-transform expression (represented by a complex number):$\begin{matrix}{{g(n)} = {\sum\limits_{k = 0}^{N - 1}{{G\left( {k/N} \right)}\exp \quad \left( {j\quad 2\pi \quad {{kn}/N}} \right)}}} & (6)\end{matrix}$

Moreover, G(n/N) and G_(n), are simplified as g(n) and g_(n),respectively, and each of the above Fourier-transform expressions isrepresented by a rotator T given by the following Expression (7):

T=exp(−j2π/N)  (7)

Fourier-transform expression (represented by a rotator): $\begin{matrix}{G_{n} = {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{g_{k}T^{nk}}}}} & (8)\end{matrix}$

Inverse Fourier-transform expression (represented by a rotator):$\begin{matrix}{g_{n} = {\sum\limits_{k = 0}^{N - 1}{G_{n}T^{- {nk}}}}} & (9)\end{matrix}$

Each of the above Fourier-transform expressions requires an enormousamount of calculation. Therefore, it is difficult to apply suchFourier-transform expressions directly to an actual operation. As aresult, more practical “fast Fourier transform” (hereinafter, simplyreferred to as “FFT”) is used.

The FFT is an algorithm wherein the number of multiplying operations ofsample signal g_(k) and rotator T^(nk) as well as the number of addingand subtracting operations represented by Σ in a transform expressionare significantly reduced.

The FFT is applied in a variety of fields, and numerous types ofalgorithms have been proposed for the FFT. Each such algorithm hasrespective specific characteristics in terms of simplicity, operationspeed, software-program configuration, advantageous property forimplementing hardware, or the like. Among these, the FFT with a radix of2 is most typically used.

The FFT with a radix of 2 is as follows:

First, it is assumed that the number N of sample values is 2^(n) (wheren is an integer). $\begin{matrix}\left. \begin{matrix}{{e(n)} = {g\left( {2n} \right)}} \\{{{{h(n)} = {g\left( {{2n} + 1} \right)}};\quad {n = 0}},1,\ldots \quad,{{N/2} - 1}}\end{matrix} \right\} & (10)\end{matrix}$

When a coefficient 1/N is omitted, the following Fourier-transformexpression can be obtained: $\begin{matrix}\begin{matrix}{G_{k} = {\sum\limits_{n = 0}^{N - 1}{g_{k}T^{nk}}}} \\{= {\sum\limits_{n = 0}^{{N/2} - 1}\left( {{{e(n)}T^{2{nk}}} + {{h(n)}T^{{({{2n} + 1})}k}}} \right)}}\end{matrix} & (11) \\{G_{k}\begin{Bmatrix}{{E_{k} + {T^{k}H_{k}}};} & {0 \leq k \leq {\frac{N}{2} - 1}} \\{{E_{k - {N/2}} + {T^{k}H_{k - {N/2}}}};} & {\frac{N}{2} \leq k \leq {N - 1}}\end{Bmatrix}} & (12) \\{where} & \quad \\\left. \begin{matrix}{E_{k} = {\sum\limits_{n = 0}^{{N/2} - 1}{{e(n)}T^{2{nk}}}}} \\{H_{k} = {\sum\limits_{n = 0}^{{N/2} - 1}{{h(n)}T^{2{nk}}}}}\end{matrix} \right\} & (13)\end{matrix}$

In the case of, for example, N=2^(n0), this algorithm can be utilized n0times, as shown in FIG. 7.

In FIG. 7, DFT indicates discrete Fourier transform. In the case of, forexample, N=2⁴, the algorithm is repeated four times.

The algorithm is primarily configured from a basic operation called“butterfly operation”. In order to implement the butterfly operation, abit-reversal method of input data and coefficient is used.

Only the fast Fourier transform has been mentioned herein. The operationof inverse Fourier transform is substantially the same as that of thefast Fourier transform, except that G_(k) and g_(n) are exchanged eachother. Therefore, description thereof is omitted.

Other specific examples include the techniques disclosed in JapaneseLaid-Open Publication Nos. 5-189470, 5-174046 and 5-189471,respectively.

Japanese Laid-Open Publication No. 5-189470 relates to a method forperforming time-series data input type Fourier transform, and disclosesFourier transform which is performed in a digital manner. In thismethod, a number of operation devices and buffers are employed togetherwith the above-mentioned FFT algorithm to perform Fourier transform Inreal time. This method is characterized in that the process is initiatedbefore all of N data have been collected.

Japanese Laid-Open Publication No. 5-174046 shows a circuitconfiguration wherein the butterfly operation is performed in a digitalmanner by using a multiplier or the like as an operation circuit.

In Japanese Laid-Open Publication No. 5-189471, a butterfly-typeoperation device performs FFT in a pipeline manner by using abit-reversal addressing technique or the like. This is a typical methodfor implementing an FFT processor.

The above-mentioned FFT algorithm is not problematic in the case ofoff-line data analysis using a high-level language. However, in the casewhere on-line data processing is conducted by using DSP (Digital SignalProcessor), that is, in the case where audio data or image data whichhas been compressed by Fourier transform is reproduced, for example, inreal time, the FFT algorithm has some disadvantages as follows:

(1) the algorithm must be changed dependent upon hardware.

Since software is dependent upon the hardware, new software and a newalgorithm must be produced when the hardware is changed, whereby thedevelopment period is increased;

(2) Since a special operation is performed, data processing other thanFFT is adversely affected.

In order to implement the bit-reversal of the butterfly operation,special addressing must be conducted by hardware. Therefore, whengeneral-purpose processes are simultaneously conducted by the samehardware, a long instruction code is required. As a result, the hardwareis not efficiently utilized, as well as an instruction-memory capacityis increased, leading to an increase in the cost;

(3) The accuracy is limited by the speed.

In order to increase the processing accuracy, the number of bits must beincreased to some extent. According to the FFT algorithm, a number ofmultiplying and adding operations are performed, whereby the speed(clock) is limited in order to assure carrier processing of suchoperations; and

(4) Power consumption is increased with an increase in clock frequency.

The on-line data processing by DSP must be conducted at a high speed. Itis a common technique to increase a clock frequency in order to performthe algorithm at a higher speed. In a digital circuit, however, powerconsumption is increased proportionally to the increase in a clockfrequency. This is not advantageous for portable equipment, since, inthe portable equipment, low power consumption is desirable in order toutilize a battery as long as possible.

For example, TMS320C50 by TEXAS INSTRUMENTS INC. requires 28,951 cyclesfor an FFT operation when the number of sample values is N=64 (whichcorresponds to 72.38 μs when a clock frequency is 40 MHz). Similarly,TMS320C50 requires 15,890 cycles when N=256, and 82,761 cycles whenN=1,024. Thus, in the case where a number of cycles are required for theoperation, a higher clock frequency must be used to increase theprocessing speed, thereby increasing the power consumption. Accordingly,the general-purpose DSP cannot be used for the portable equipment.

Since each of Japanese Laid-Open Publication Nos. 5-189470, 5-174046 and5-189471 utilizes a digital processor dedicated to FFT, the sameproblems as those of the general-purpose DSP arise.

SUMMARY OF THE INVENTION

The Fourier transform apparatus according to the present inventionincludes: a signal generating section for generating a plurality ofsine-wave signals and a plurality of cosine-wave signals; a plurality ofanalog circuits each having a respective circuit parameter correspondingto a respective Fourier coefficient, and each receiving the respectivesine-wave signal and the respective cosine-wave signal which aregenerated by the signal generating section; and an operation section forperforming an operation on each of outputs of the respective analogcircuits and outputting the resultant respective analog signals.

With such a configuration, the signal generating section generates thesine-wave signals and cosine-wave signals, and inputs the sine-wavesignals and cosine-wave signals to the respective analog circuits. Eachof the analog circuits has a respective circuit parameter for Fourierseries, and performs an operation of the respective sine-wave signal andthe respective cosine-wave signal based on the respective Fourierseries. The operation section performs an operation on each of theoutputs of the respective analog circuits and outputs the resultantrespective analog signals.

Provided that the sine-wave signals and the cosine-wave signals aresine-wave components and cosine-wave components of Expression (3),respectively, each analog signal output from the operation section is asignal g(n) in Expression (3).

In other words, according to the present invention, analog circuitry isemployed at least partially in the Fourier transform apparatus, andinverse Fourier transform as defined by the following Expression (3) isperformed: $\begin{matrix}{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}{{a\left( {k/N} \right)}\cos \quad \left( {2\pi \quad {{kn}/N}} \right)}} + {\sum\limits_{k = 1}^{N - 1}{{b\left( {k/N} \right)}\sin \quad \left( {2\pi \quad {{kn}/N}} \right)}}}} & (3)\end{matrix}$

In one example, the signal generating section generates a direct-currentsignal along with the sine-wave signals and the cosine-wave signals;each of the analog circuits has a respective circuit parametercorresponding to a respective Fourier coefficient, and receives thedirect-current signal, the respective sine-wave signal and therespective cosine-wave signal which are generated by the signalgenerating section; and the operation section performs an operation oneach of the outputs of the respective analog circuits and outputs theresultant respective analog signals.

Herein, a direct-current component in the above Expression (3) is alsosubjected to processing.

In one example, the signal generating section includes a discrete signalprocessing circuit. For example, the signal generating section includes:a storing section for storing respective values of a plurality of pointson a single cycle of a sine wave; a converting section for convertingthe respective values of the points stored in the storing section torespective signals; a holding section for holding the signals convertedby the converting section: and a signal forming section for sequentiallyoutputting the signals held by the holding section during respectivedistinct cycles, thereby generating the sine-wave signals andcosine-wave signals having respective cycles.

In the case where the signal generating section is a discrete signalprocessing circuit, a variety of sine-wave signals and cosine-wavesignals can be reproduced with a high accuracy by the discrete signalprocessing circuit.

Moreover, since each of the analog circuits immediately generates arespective output for the respective input (i.e., respective sine-wavesignal, cosine-wave signal and direct current), advantages of thediscrete signal processing circuit (i.e., high accuracy and flexibility)as well as advantages of the analog circuits (i.e., high-speedprocessing) can be sufficiently achieved.

In one example, the signal generating section generates a direct-currentsignal corresponding to constant 1 in a first term of a right side ofExpression (3), the plurality of cosine-wave signals corresponding tocos(2πkn/N) in a second term of the right side of the Expression (3),and the plurality of sine-wave signals corresponding to sin(2πkn/N) in athird term of the right side of the Expression (3): $\begin{matrix}{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}{{a\left( {k/N} \right)}\cos \quad \left( {2\pi \quad {{kn}/N}} \right)}} + {\sum\limits_{k = 1}^{N - 1}{{b\left( {k/N} \right)}\sin \quad {\left( {2\pi \quad {{kn}/N}} \right).}}}}} & (3)\end{matrix}$

In one example, each of the analog circuits has a circuit parametercorresponding to a Fourier coefficient a(0) in a first term of a rightside of Expression (3), a plurality of circuit parameters correspondingto a Fourier coefficient a(k/N) in a second term of the right side ofthe Expression (3), and a plurality of circuit parameters correspondingto a Fourier coefficient b(k−N) in a third term of the right side of theExpression (3): $\begin{matrix}{{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{N - 1}\quad {{b\left( {k/N} \right)}{{\sin \left( {2\pi \quad k\quad {n/N}} \right)}.}}}}}} & (3)\end{matrix}$

In one example, the operation section adds the respective outputs of theanalog circuits, thereby outputting an analog signal corresponding tog(n) of a left side of Expression (3): $\begin{matrix}{{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{N - 1}\quad {{b\left( {k/N} \right)}{{\sin \left( {2\pi \quad k\quad {n/N}} \right)}.}}}}}} & (3)\end{matrix}$

In one example, each of the circuit parameters of the analog circuits isa resistance value.

In one example, each of the respective resistances of the analogcircuits is a variable resistance, and the Fourier transform apparatusfurther includes a changing section for changing the variableresistances of the respective analog circuits.

Herein, the respective circuit parameters of the analog circuits, thatis, the resistances, can be readily changed by the changing section,thereby achieving excellent flexibility.

Thus, the invention described herein makes possible the advantage ofproviding a Fourier transform apparatus having a reduced number ofoperations as well as a Fourier transform apparatus capable of rapidlyand accurately performing the operations, thereby achieving low powerconsumption.

This and other advantages of the present invention will become apparentto those skilled in the art upon reading and understanding the followingdetailed description with reference to the accompanying figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a Fourier transfer apparatus according toone example of the present invention;

FIG. 2 is a signal timing chart illustrating an operation of theapparatus shown in FIG. 1;

FIG. 3 is a signal timing chart illustrating another operation of theapparatus shown in FIG. 1;

FIG. 4 is a block diagram of a configuration of a sine/cosine-wavesignal generating section in the apparatus shown in FIG. 1;

FIG. 5 is a block diagram showing one example of variable resistancesections of a Fourier coefficient transform section in the apparatusshown in FIG. 1;

FIG. 6 is a block diagram showing another example of the variableresistance sections of the Fourier coefficient transform section in theapparatus shown in FIG. 1; and

FIG. 7 is a schematic block diagram of a conventional apparatus forperforming an FFT operation.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, the present invention will be described with reference tothe accompanying drawings.

FIG. 1 shows a Fourier transform apparatus according to one example ofthe present invention. The Fourier transform apparatus of the presentexample conducts a process corresponding to inverse Fourier transformgiven by the above Expression (3), and includes a sine/cosine-wavesignal generating section 1, a Fourier coefficient transform section 2and an adding section 3.

The sine/cosine-wave signal generating section 1 generates an analogdirect-current signal and (2N−2) analog sine/cosine-wave signalssin(ωt), cos(ωt), . . . , and outputs these signals.

The Fourier coefficient transform section 2 includes variable resistancesections r_(r0), r_(i1), r_(r1), r_(i2), . . . and r_(r(N−1))corresponding to respective Fourier coefficients, and changes aresistance of the respective variable resistance sections in response toa change of the respective Fourier coefficients.

The adding section 3 includes an adding circuit 3-1 and a resistorr_(f). The adding section 3 adds the respective outputs of the variableresistor sections corresponding to the respective Fourier coefficientsin the Fourier coefficient transform section 2, and outputs an analogsignal indicating the addition result.

The Fourier transform apparatus according to the present examplegenerates an analog direct-current signal and analog sine/cosine-wavesignals, and changes a resistance of the respective variable resistancesections in the Fourier coefficient transform section 2 in a stepwisemanner, that is, changes the respective Fourier coefficient in a digitalmanner. Accordingly, this Fourier transform apparatus is herein referredto as an analog/digital-mixed circuit.

It is now assumed that a main frequency f of a sine/cosine-wave signalis equal to a fundamental-wave frequency f_(N) of a sample signal, asgiven by the following Expression (14): $\begin{matrix}\left. \begin{matrix}{f = f_{N}} \\{\omega = {2\pi \quad f}}\end{matrix} \right\} & (14)\end{matrix}$

In this case, as shown in the timing chart of FIG. 2, at time t=0, thesine/cosine-wave signal generating section 1 generates a direct-currentsignal and sine/cosine-wave signals while the Fourier coefficienttransform section 2 sets the resistance values of the respectivevariable resistance sections corresponding to the respective Fouriercoefficients. Thereafter, the Fourier coefficient transform section 2sequentially updates these respective resistance values in everyprescribed cycle T=1/f. Thus, the Fourier transform apparatuscontinuously performs inverse Fourier transform, and outputs a signalV₀(t) from the adding section 3, as given by the following Expression(15): $\begin{matrix}{{V_{0}(t)} = {R_{f}\left\{ {\frac{1}{R_{r\quad 0}\left( a_{0} \right)} + {\sum\limits_{n = 1}^{N - 1}\quad {\frac{1}{R_{m}\left( a_{n} \right)}{\cos \left( {n\quad \omega \quad t} \right)}}} + {\sum\limits_{n = 1}^{N - 1}\quad {\frac{1}{R_{in}\left( b_{n} \right)}{\sin \left( {n\quad \omega \quad t} \right)}}}} \right\}}} & (15)\end{matrix}$

where R_(f) is a resistance value of the resistor r_(f), and R_(r0),R_(i0), R_(r1), R_(i12), . . . , and R_(r1(N−1)) are resistance valuesof the respective variable resistance sections r_(r0), r_(i0), r_(r1),r_(i12), . . . , and r_(r1(N−1)).

As can be seen from the comparison between the above Expressions (3) and(15), the Fourier coefficients a, a/(k/N) and b(k/N) correspond toR_(f)/R_(r0), R_(f)/R_(rn) and R_(f)/R_(in), respectively. Therefore, itcan be said that the Fourier transform apparatus of the present exampleperforms inverse Fourier transform.

Also, as indicated by the following Expression (16), it is hereinassumed that a main frequency f of each sine/cosine-wave signal ishigher than the fundamental-wave frequency f_(N) of the sample signal asgiven by Expression (1): $\begin{matrix}\left. \begin{matrix}{f > f_{N}} \\{\omega = {2\pi \quad f}}\end{matrix} \right\} & (16)\end{matrix}$

In this case, as shown in the timing chart of FIG. 3, eachsine/cosine-wave signal generated by the sine/cosine-wave signalgenerating section 1 must be reset at every interval T. The outputsignal V₀(t) of the adding section 3 is valid only during a time periodt=iT to (iT−ΔT), as defined by the following Expression (17), where iTis a start time of each cycle T. Thus, another process can be performedduring the time period ΔT. $\begin{matrix}{{{{V_{0}(t)} = {R_{f}\left\{ {\frac{1}{R_{r\quad 0}\left( a_{0} \right)} + {\sum\limits_{n = 1}^{N - 1}\quad {\frac{1}{R_{m}\left( a_{n} \right)}{\cos \left( {n\quad \omega \quad t} \right)}}} + {\sum\limits_{n = 1}^{N - 1}\quad {\frac{1}{R_{in}\left( b_{n} \right)}{\sin \left( {n\quad \omega \quad t} \right)}}}} \right\}}};}{{{{\left( {i - 1} \right)T} \leq t \leq {{i\quad T} - {\Delta \quad T}}};{i = 0}},1,\ldots}} & (17)\end{matrix}$

Whichever of Expressions (15) and (17) is used, the signal V₀ resultingfrom the Fourier transform may be used either directly as an analogsignal or used after being converted into a digital signal.

FIG. 4 is a block diagram showing a specific example of thesine/cosine-wave signal generating section 1. In this case, a Shannonsampling theorem is applied, thereby reducing the number ofsine/cosine-wave signals required for inverse Fourier transform.Accordingly, the number of variable resistance sections of the Fouriertransform section 2 is also reduced.

According to the Shannon sampling theorem, a signal frequency which canbe restored is equal to or less than the half of a sampling frequency.The above Expression (3) can be rewritten to the following Expression(18), wherein the upper limit of a variable n of Expression (3) isreduced from N−1 to N/2−1. $\begin{matrix}{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{{N/2} - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{{N/2} - 1}\quad {{b\left( {k/N} \right)}{\sin \left( {2\pi \quad k\quad {n/N}} \right)}}}}} & (18)\end{matrix}$

The sine/cosine-wave signal generating section 1 stores respectivevoltage values at N point(s) on a sine-wave in a ROM (read only memory)11. It should be noted that the number of points N is desirably 2^(n)(where n is an integer). In such a case, the respective voltage valuesat the N points become symmetric to each other with respect to the peakof the sine-wave (i.e., the sine wave has the same value on both sidesof the peak), whereby the values for the half of each cycle of thesine-wave can be omitted. Therefore, the capacity of the ROM 11 as wellas the number of sample holding circuits 14 can be reduced by half.

A sine/cosine value corresponding to the remainder of kn divided by Nwhen kn is out of the range of 0 to (N−1) is defined by the followingExpression (19):

sin(2πkn/N)∈{sin(2πm/N)};

cos(2πkn/N)∈{cos(2πm/N};  (19)

k, n, m=0, 1, . . . , N−1

The trigonometric function has the following relation:

cos(ωt)=sin(ωt+π/2)  (20)

A digital/analog converter (DAC) 12 extracts N values on a single cycleof the sine-wave from the sine/cosine-wave signal generating section 1,and converts the extracted values to respective analog signals foroutput. These analog signals are distributed through an analogdemultiplexer 13 to N sample holding circuits 14, respectively.

N/2−1 multiplexers 15 sequentially extract the respective analog signalsfrom the respective sample holding circuits 14 in a preset order so asto output, at a respective timing, the voltage values of the respectivesine-waves having the respective cycles, thereby generating N/2−1sine-wave signals. Moreover, the other N/2−1 multiplexers 15 extract therespective analog signals from the respective sample holding circuits 15and output the extracted analog signals in the respective cycles,thereby generating N/2−1 cosine-wave signals.

Thus, the multiplexers 15 sequentially extract the analog signals fromthe respective sample holding circuits 14 and output the extractedsignals, thereby producing 2(N/2−1) sine-wave signals and 2(N/2−1)cosine-wave signals.

In the sine/cosine-wave signal generating section 1 as shown in FIG. 4,a frequency of the sine/cosine-wave signal is not particularly limited.A direct current signal is separately produced for output. S1 indicatesa synchronization signal for refreshing the sample holding circuits 14.S2 indicates a synchronization signal of the multiplexers 15.

The synchronization signal S2 is synchronized with the fundamental-wavefrequency f_(N) of the sample signal as given by Expression (1) so as tomake the fundamental-wave frequency f_(N) equal to the main frequency fof the sine/cosine wave signal. In this case, as shown in the timingchart of FIG. 2, the direct-current signal and sine/cosine-wave signalsare generated by the sine/cosine-wave signal generating section 1.

Moreover, the cycle of the synchronization signal S2 is reduced so as tomake the main frequency f of the sine/cosine-wave signal shorter thanthe fundamental-wave frequency f_(N) of the sample signal. In this case,as shown in the timing chart of FIG. 3, the output signal v₀(t) of theadding section 3 is valid only during a time period t=iT to (iT−ΔT),where iT is a start time of each cycle T. Accordingly, in the case wherethe output signal V₀(t) of the adding section 3 is received by areceiving section (not shown) only during the time period of iT to(iT−ΔT) in synchronization with the synchronization signal S2, a signalhaving the same frequency spectrum as that of the sample signal can berestored. As a result, the receiving section can carry out anotherprocess during the time period ΔT.

In the above example, each sine/cosine-wave signal is output in adiverse manner in synchronization with the timing signal S2. However,the present invention can be implemented even when a sine/cosine wavesignal as a continuous analog waveform is used.

FIG. 5 is a block diagram showing a specific example of the variableresistance sections of the Fourier coefficient transform section 2.

Each variable resistance section is of a current-adding type, andincludes a single resistor r₀, r₁, . . . , r_((l−1)) and a single switchS₀. Each switch S₀ receives a respective 1-bit Q_(l−1), Q_(l−2), . . . ,Q₀, and is turned ON or OFF depending upon the received bit value. Byselectively turning ON the switches S₀, resistance values of therespective variable resistance sections corresponding to the respectiveFourier coefficient Q_(m) are set.

When 1-bit of data representing the respective Fourier coefficient Qm isapplied to the respective variable resistance sections, the relationamong the respective resistance values 2⁰R0, 2¹R0, . . . , 2^((l−1))R0,the Fourier coefficient Qm and their equivalent resistances is given bythe following Expression (21): $\begin{matrix}{R_{e} = {R_{0}/{\sum\limits_{m = 0}^{l - 1}\quad {2^{m - {({l - 1})}}Q_{m}}}}} & (21)\end{matrix}$

where Qm is a Fourier coefficient, and m=0, 1, . . . , l−1.

FIG. 6 is a block diagram showing another specific example of thevariable resistance sections in the Fourier coefficient transformsection 2.

Each variable resistance section is of a voltage-adding type, andincludes a single resistor r₀, (l+1) resistors 2r₀, and a single switchS₀. Each switch S₀ is turned ON or OFF in response to respective 1-bitQ_(l−1), Q_(l−2), . . . , Q₀ which indicates a respective Fouriercoefficient Q_(m). By selectively turning ON the switches S₀, resistancevalues of the respective variable resistance sections corresponding tothe respective Fourier coefficient Q_(m) are set.

When 1-bit of data representing the respective Fourier coefficient Q_(m)is applied to the respective variable resistance sections, the relationamong the resistance values R₀ and 2R₀ (R₀=2R₀/2) of the respectiveresistors r₀ and 2r₀, the Fourier coefficient Q_(m) and their equivalentresistances is given by the following Expression (22): $\begin{matrix}{R_{e} = {6{R_{0}/{\sum\limits_{m = 0}^{l - 1}\quad {2^{m - {({l - 1})}}Q_{m}}}}}} & (22)\end{matrix}$

where Q_(m) is a Fourier coefficient, and m=0, 1, . . . , l−1.

It should be noted that the present invention is not limited to theabove-described example, and various modifications can be made to thepresent invention. For example, in place of the variable resistancesections of the Fourier coefficient transform section 2, other types ofcircuits (e.g., circuits for converting a Fourier coefficient to acapacitance by capacitive coupling) may be provided. However, in thecase where a capacitance and/or a inductance are used, high-speedoperation and high accuracy cannot be sufficiently realized due to thelarge inertia (such as residual voltage). Therefore, it is desirable touse a variable resistance.

As can be seen from the foregoing, according to the present invention,analog circuitry is employed at least partially in the Fourier transformapparatus. Therefore, the inverse Fourier transform as defined by theabove Expression (3) can be performed.

With such a configuration, the Fourier transform apparatus does not relyon software. Therefore, the Fourier transform can be performed only bydedicated hardware, eliminating the need for software development. Thus,the development period is not increased in this respect.

Moreover, since no special operation is performed, the cost of theapparatus can be reduced.

Furthermore, since the operation is not repeated, a clock frequency neednot be increased in order to improve the processing accuracy, wherebythe power consumption is not increased.

According to the present invention, when the number of sample values isN=64, the number of cycles required for the FFT operation is 64.Similarly, the number of cycles required for the FFT operation is 256when N=256, and 1,024 when N=1,024. This indicates that the requirednumber of cycles in the present invention is reduced to the range of{fraction (1/45)} to {fraction (1/80)} of the required number of cyclesin the above-mentioned general-purpose DSP. Therefore, the powerconsumption can be significantly reduced.

Various other modifications will be apparent to and can be readily madeby those skilled in the art without departing from the scope and spiritof this invention. Accordingly, it is not intended that the scope of theclaims appended hereto be limited to the description as set forthherein, but rather that the claims be broadly construed.

What is claimed is:
 1. A Fourier transform apparatus, comprising: asignal generating section for generating a plurality of sine-wavesignals and a plurality of cosine-wave signals; a plurality of analogcircuits each having a respective circuit parameter corresponding to arespective Fourier coefficient, and each receiving the respectivesine-wave signal and the respective cosine-wave signal which aregenerated by the signal generating section; and an operation section forperforming an operation on each of outputs of the respective analogcircuits and outputting the resultant respective analog signals.
 2. AFourier transform apparatus according to claim 1, wherein the signalgenerating section generates a direct-current signal along with thesine-wave signals and the cosine-wave signals, and each of the analogcircuits has a respective circuit parameter corresponding to arespective Fourier coefficient, and receives the direct-current signal,the respective sine-wave signal and the respective cosine-wave signalwhich are generated by the signal generating section; and the operationsection performs an operation on each of the outputs of the respectiveanalog circuits and outputting the resultant respective analog signals.3. A Fourier transform apparatus according to claim 2, wherein thesignal generating section generates the direct-current signalcorresponding to constant 1 in a first term of a right side ofExpression (3), the plurality of cosine-wave signals corresponding tocos(2πkn/N) in a second term of the right side of the Expression (3),and the plurality of sine-wave signals corresponding to sin(2πkn/N) in athird term of the right side of the Expression (3): $\begin{matrix}{{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{N - 1}\quad {{b\left( {k/N} \right)}{{\sin \left( {2\pi \quad k\quad {n/N}} \right)}.}}}}}} & (3)\end{matrix}$


4. A Fourier transform apparatus according to claim 2, wherein each ofthe analog circuits has a circuit parameter corresponding to a Fouriercoefficient a(0) in a first term of a right side of Expression (3), aplurality of circuit parameters corresponding to a Fourier coefficienta(k/N) in a second term of the right side of the Expression (3), and aplurality of circuit parameters corresponding to a Fourier coefficientb(k/N) in a third term of the right side of the Expression (3):$\begin{matrix}{{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{N - 1}\quad {{b\left( {k/N} \right)}{{\sin \left( {2\pi \quad k\quad {n/N}} \right)}.}}}}}} & (3)\end{matrix}$


5. A Fourier transform apparatus according to claim 2, wherein theoperation section adds the respective outputs of the analog circuits,thereby outputting an analog signal corresponding to g(n) of a left sideof Expression (3): $\begin{matrix}{{{g(n)} = {{a(0)} + {\sum\limits_{k = 1}^{N - 1}\quad {{a\left( {k/N} \right)}{\cos \left( {2\pi \quad k\quad {n/N}} \right)}}} + {\sum\limits_{k = 1}^{N - 1}\quad {{b\left( {k/N} \right)}{{\sin \left( {2\pi \quad k\quad {n/N}} \right)}.}}}}}} & (3)\end{matrix}$


6. A Fourier transform apparatus according to claim 1, wherein thesignal generating section includes a discrete signal processing circuit.7. A Fourier transform apparatus according to claim 6, wherein thesignal generating section includes a storing section for storingrespective values of a plurality of points on a single cycle of a sinewave; a converting section for converting the respective values of thepoints stored in the storing section to respective signals; a holdingsection for holding the signals converted by the converting section; anda signal forming section for sequentially outputting the signals held bythe holding section during respective distinct cycles, therebygenerating the sine-wave signals and cosine-wave signals havingrespective cycles.
 8. A Fourier transform apparatus according to claim1, wherein each of the circuit parameters of the analog circuits is aresistance value.
 9. A Fourier transform apparatus according to claim 8,wherein each of the respective resistances of the analog circuits is avariable resistance, the Fourier transform apparatus further comprising:a changing section for changing the variable resistances of therespective analog circuits.